Integral (Power Rule) Calculator Formula
Understand the math behind the integral (power rule) calculator. Each variable explained with a worked example.
Formulas Used
New Coeff
new_coeff = n != -1 ? coeff / (n + 1) : 0New Exp
new_exp = n + 1Definite
definite = n != -1 ? (coeff / (n + 1)) * (pow(upper, n + 1) - pow(lower, n + 1)) : 0Variables
| Variable | Description | Default |
|---|---|---|
coeff | Coefficient (a) | 6 |
n | Exponent (n) | 2 |
lower | Lower Bound | 0 |
upper | Upper Bound | 3 |
How It Works
Power Rule for Integration
Rule
integral of ax^n dx = (a/(n+1)) × x^(n+1) + C (when n ≠ -1)
For n = -1, the integral is a × ln
Definite Integral
Evaluate the antiderivative at the upper and lower bounds and subtract:
F(upper) - F(lower)
Worked Example
Integrate 6x² from 0 to 3.
- 01Antiderivative: (6/3)x³ = 2x³
- 02F(3) = 2 × 27 = 54
- 03F(0) = 2 × 0 = 0
- 04Definite integral = 54 - 0 = 54
Frequently Asked Questions
What is the power rule for integration?
The integral of x^n is x^(n+1)/(n+1) + C, valid for all n except -1. This reverses the power rule for derivatives.
What is + C in indefinite integrals?
C is the constant of integration. Since the derivative of any constant is 0, the antiderivative includes an unknown constant.
What happens when n = -1?
When n = -1, the integral of 1/x is ln|x| + C (the natural logarithm). The power rule formula does not apply because it would involve division by zero.
Learn More
Guide
Introduction to Derivatives - Complete Guide
Learn the fundamentals of derivatives in calculus. Covers the definition, power rule, product rule, chain rule, and practical applications with clear examples.
Ready to run the numbers?
Open Integral (Power Rule) Calculator